The hydrogen atom in a space with one compact extra dimension: explicit and computer-assisted bounds on the ground-state energy up to and including the critical radius

Abstract. We study the non-relativistic one-electron atom with a point nucleus in the space ℝ³ × S¹, where the circle has radius R and the potential is the one given by Gauss' law in four space dimensions, periodised over the circle; this is the model of Bureš and Siegl (2015). In units with Bohr radius a₀ = 1 and energies in Rydberg, let E₀(R) be the bottom of the spectrum of the Friedrichs extension of the minimal operator. We prove the following. (i) For 0 < R ≤ a₀/4, the critical radius a₀/4 included, E₀(R) ≥ −4/(1+√(1−4R))² ≥ −4; the proof also gives a Hardy-type inequality for the periodised inverse-square potential, with constant 1 and an explicit remainder. (ii) For 0 < R ≤ a₀/20, −43R⁴ ≤ E₀(R)+1+4ζ(3)R³ ≤ 9R⁴; in particular the shift from the hydrogen value −1 is of order R³. (iii) At the five radii R = 1/20, 1/10, 3/20, 1/5, 1/4 we give computer-assisted two-sided bounds; at the critical radius, −1.2998 ≤ E₀(1/4) ≤ −1.29405. We also show that for 0 < R ≤ a₀/4 the essential spectrum is [0,∞) and E₀(R) is an eigenvalue, so that the numbers above are bounds on the ground-state energy; for R < a₀/4 this follows from what the prior work states; at the endpoint our self-contained proof uses standard methods, and we sketch, without using it, how the result can also be obtained, in our reading, from known theorems. With two or more compact dimensions the quadratic form is unbounded below, by a standard scaling argument. Formulas and values printed in earlier work are compared with these results as statements of fact only. We make no statement on whether extra dimensions exist. The statements, proofs and computer certificates were written by an AI system and reviewed by independent instances of AI systems; they have not been refereed by humans. Version 1.0 (13 October 2026). This record holds the paper (PDF) and an archive of the project repository without its version history: the computer certificates, the proof notes, the review findings, the records and the tools (see README.md; most project documents are in Japanese, and a Japanese summary of the paper is in the archive). Contact: [email protected].